Structure preserving transformations

Deriving the generating functions

A final addendum, for now, in the series on mathematical economics. I had never actually derived the dynamic formulas for the generating function, or argued as to why it should exit. This post scratches that itch, arguing along the lines of physicist S.G.Rajeev here (arXiv).

Remember we are looking for transformations that preserve equilibrium manifolds, and those are ones where we can find a utility function s=U(q) so that ds = pᵢdqⁱ. In certain coordinates. We can change these coordinates with a transformation, but we want find transformations that preserve the relationship when s→sʹ, q→qʹ and p→pʹ. So

\(ds' - p'dq' = \lambda (ds-pdq) \qquad \qquad (1)\)

because if the right is 0, so is the left (for any λ). We ignore the indices for convenience. Transformations with this property are called Legendre transformations.

Now express the new coordinates as functions of the old ones. We have

\(ds' = \frac{\partial s'}{\partial s}ds + \frac{\partial s'}{\partial q}dq + \frac{\partial s'}{\partial p}dp\)

and similar for dqʹ and dpʹ. That gives us, using (1), that (deep breath)

\(\frac{\partial s'}{\partial p} - p'\frac{\partial q'}{\partial p}=0, \qquad \frac{\partial s'}{\partial s} - p'\frac{\partial q'}{\partial s}=\lambda, \qquad \frac{\partial s'}{\partial q} - p'\frac{\partial q'}{\partial q}=-\left( \frac{\partial s'}{\partial s} - p'\frac{\partial q'}{\partial s} \right)p \qquad (2)\)

That is, respectively: the coefficient in (1) for dp is 0, for ds is λ, and for dq is -λp.

We now consider this transformation infinitesimally, i.e.

\(s \to s'= s+tX_s, \quad q \to q'= q+tX_q, \quad p \to p' = p + tX_p, \quad |t| \ll 1\)

for some vector X = Xₛ∂ₛ + X₉∂₉ + Xₚ∂ₚ. Defining

\(G = pX_q - X_s\)

we can observe that:

\(X_q = \frac{\partial G}{\partial p}, \qquad X_s = p\frac{\partial G}{\partial p} -G, \qquad X_p = -\frac{\partial G}{\partial q} -p\frac{\partial G}{\partial s},\)

The first two equations follow directly from the definition of G, the third by expanding the rightmost of (2) in the first order of t:

\(\frac{\partial s'}{\partial q} - p'\frac{\partial q'}{\partial q} = -X_p - \frac{\partial G}{\partial q} \qquad -\left( \frac{\partial s'}{\partial s} - p'\frac{\partial q'}{\partial s} \right)p =\frac{\partial G}{\partial s}p\)

and equating the two as in (1) leads to the third identity¹. We can turn this around and note that for any function G(s,q,p) we can define an infinitesimal transformation

\(X_G = \left(\frac{\partial G}{\partial p}\right)\partial_q + \left(-\frac{\partial G}{\partial q}-p\frac{\partial G}{\partial s}\right)\partial_p + \left(p\frac{\partial G}{\partial p}-G\right)\partial_s \qquad (3)\)

that will preserve the structure!

There is a coordinate-free argument involving the Lie derivative to introduce X_G, which I will try to reproduce². It is that an infinitesimal transformation X preserves the structure α if the Lie derivative scales it (where the right is Cartan’s magic formula)

\(\mathcal{L}_X \alpha = \lambda \cdot \alpha = i_X d\alpha + di_X \alpha\)

We associate a vector field X_G to a function G by requiring

\(i_{X_G} \alpha =\alpha(X_G) = G \quad \text{and} \quad i_{X_G} d\alpha = -DG = \frac{\partial G}{\partial s}\alpha - dG\)

where I have cheated slightly by writing ∂ₛ for the characteristic (or Reeb) vector field ξ, characterized by i_ξ dα = 0 and i_ξ α = α(ξ) = 1. The covariant differential D is defined as the “horizontal” part of the differential so Df ≝ df - ξ(f)α. Comparing the above with the Lie derivative, we see we must have λ = ∂G/∂s and other dynamics in (3) follow if we choose coordinates to express α = ds - pdq.

For earlier posts in this series:

Mathematical economics (reference)

1. If you think I find this easy, I just spent an hour hunting for a minus sign.

2. This is even less easy, I am proud I can even still read the original math articles.